X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Flibrary%2FZ%2Fsigma_p.ma;h=92b21539605e564044f85b67228b61d0077b627f;hb=35525e5acd7854210e2a1bbba07a4909117029ac;hp=73b29a38c0773b14f86ee84f07403a2419441498;hpb=efce71f56f343e316b2cd4834cbf3f5dcb3571eb;p=helm.git

diff --git a/matita/library/Z/sigma_p.ma b/matita/library/Z/sigma_p.ma
index 73b29a38c..92b215396 100644
--- a/matita/library/Z/sigma_p.ma
+++ b/matita/library/Z/sigma_p.ma
@@ -12,34 +12,41 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/Z/sigma_p.ma".
+set "baseuri" "cic:/matita/Z/sigma_p".
 
 include "Z/times.ma".
 include "nat/primes.ma".
 include "nat/ord.ma".
+include "nat/generic_iter_p.ma".
 
-let rec sigma_p n p (g:nat \to Z) \def
-  match n with
-   [ O \Rightarrow OZ
-   | (S k) \Rightarrow 
-      match p k with
-      [true \Rightarrow (g k)+(sigma_p k p g)
-      |false \Rightarrow sigma_p k p g]
-   ].
+(* sigma_p in Z is a specialization of iter_p_gen *)
+definition sigma_p: nat \to (nat \to bool) \to (nat \to Z) \to Z \def
+\lambda n, p, g. (iter_p_gen n p Z g OZ Zplus).
 
+theorem symmetricZPlus: symmetric Z Zplus.
+change with (\forall a,b:Z. (Zplus a b) = (Zplus b a)).
+intros.
+rewrite > sym_Zplus.
+reflexivity.
+qed.
+   
 theorem true_to_sigma_p_Sn: 
 \forall n:nat. \forall p:nat \to bool. \forall g:nat \to Z.
 p n = true \to sigma_p (S n) p g = 
 (g n)+(sigma_p n p g).
-intros.simplify.
-rewrite > H.reflexivity.
+intros.
+unfold sigma_p.
+apply true_to_iter_p_gen_Sn.
+assumption.
 qed.
    
 theorem false_to_sigma_p_Sn: 
 \forall n:nat. \forall p:nat \to bool. \forall g:nat \to Z.
 p n = false \to sigma_p (S n) p g = sigma_p n p g.
-intros.simplify.
-rewrite > H.reflexivity.
+intros.
+unfold sigma_p.
+apply false_to_iter_p_gen_Sn.
+assumption.
 qed.
 
 theorem eq_sigma_p: \forall p1,p2:nat \to bool.
@@ -47,32 +54,10 @@ theorem eq_sigma_p: \forall p1,p2:nat \to bool.
 (\forall x.  x < n \to p1 x = p2 x) \to
 (\forall x.  x < n \to g1 x = g2 x) \to
 sigma_p n p1 g1 = sigma_p n p2 g2.
-intros 5.elim n
-  [reflexivity
-  |apply (bool_elim ? (p1 n1))
-    [intro.
-     rewrite > (true_to_sigma_p_Sn ? ? ? H3).
-     rewrite > true_to_sigma_p_Sn
-      [apply eq_f2
-        [apply H2.apply le_n.
-        |apply H
-          [intros.apply H1.apply le_S.assumption
-          |intros.apply H2.apply le_S.assumption
-          ]
-        ]
-      |rewrite < H3.apply sym_eq.apply H1.apply le_n
-      ]
-    |intro.
-     rewrite > (false_to_sigma_p_Sn ? ? ? H3).
-     rewrite > false_to_sigma_p_Sn
-      [apply H
-        [intros.apply H1.apply le_S.assumption
-        |intros.apply H2.apply le_S.assumption
-        ]
-      |rewrite < H3.apply sym_eq.apply H1.apply le_n
-      ]
-    ]
-  ]
+intros.
+unfold sigma_p.
+apply eq_iter_p_gen;
+  assumption.
 qed.
 
 theorem eq_sigma_p1: \forall p1,p2:nat \to bool.
@@ -80,42 +65,17 @@ theorem eq_sigma_p1: \forall p1,p2:nat \to bool.
 (\forall x.  x < n \to p1 x = p2 x) \to
 (\forall x.  x < n \to p1 x = true \to g1 x = g2 x) \to
 sigma_p n p1 g1 = sigma_p n p2 g2.
-intros 5.
-elim n
-  [reflexivity
-  |apply (bool_elim ? (p1 n1))
-    [intro.
-     rewrite > (true_to_sigma_p_Sn ? ? ? H3).
-     rewrite > true_to_sigma_p_Sn
-      [apply eq_f2
-        [apply H2
-          [apply le_n|assumption]
-        |apply H
-          [intros.apply H1.apply le_S.assumption
-          |intros.apply H2
-            [apply le_S.assumption|assumption]
-          ]
-        ]
-      |rewrite < H3.apply sym_eq.apply H1.apply le_n
-      ]
-    |intro.
-     rewrite > (false_to_sigma_p_Sn ? ? ? H3).
-     rewrite > false_to_sigma_p_Sn
-      [apply H
-        [intros.apply H1.apply le_S.assumption
-        |intros.apply H2
-          [apply le_S.assumption|assumption]
-        ]
-      |rewrite < H3.apply sym_eq.apply H1.apply le_n
-      ]
-    ]
-  ]
+intros.
+unfold sigma_p.
+apply eq_iter_p_gen1;
+  assumption.
 qed.
 
 theorem sigma_p_false: 
 \forall g: nat \to Z.\forall n.sigma_p n (\lambda x.false) g = O.
 intros.
-elim n[reflexivity|simplify.assumption]
+unfold sigma_p.
+apply iter_p_gen_false.
 qed.
 
 theorem sigma_p_plus: \forall n,k:nat.\forall p:nat \to bool.
@@ -123,41 +83,23 @@ theorem sigma_p_plus: \forall n,k:nat.\forall p:nat \to bool.
 sigma_p (k+n) p g 
 = sigma_p k (\lambda x.p (x+n)) (\lambda x.g (x+n)) + sigma_p n p g.
 intros.
-elim k
-  [reflexivity
-  |apply (bool_elim ? (p (n1+n)))
-    [intro.
-     simplify in \vdash (? ? (? % ? ?) ?).    
-     rewrite > (true_to_sigma_p_Sn ? ? ? H1).
-     rewrite > (true_to_sigma_p_Sn n1 (\lambda x.p (x+n)) ? H1).
-     rewrite > assoc_Zplus.
-     rewrite < H.reflexivity
-    |intro.
-     simplify in \vdash (? ? (? % ? ?) ?).    
-     rewrite > (false_to_sigma_p_Sn ? ? ? H1).
-     rewrite > (false_to_sigma_p_Sn n1 (\lambda x.p (x+n)) ? H1).
-     assumption.
-    ]
-  ]
+unfold sigma_p.
+apply (iter_p_gen_plusA Z n k p g OZ Zplus)
+[ apply symmetricZPlus.
+| intros.
+  apply cic:/matita/Z/plus/Zplus_z_OZ.con
+| apply associative_Zplus
+]
 qed.
 
 theorem false_to_eq_sigma_p: \forall n,m:nat.n \le m \to
 \forall p:nat \to bool.
 \forall g: nat \to Z. (\forall i:nat. n \le i \to i < m \to
 p i = false) \to sigma_p m p g = sigma_p n p g.
-intros 5.
-elim H
-  [reflexivity
-  |simplify.
-   rewrite > H3
-    [simplify.
-     apply H2.
-     intros.
-     apply H3[apply H4|apply le_S.assumption]
-    |assumption
-    |apply le_n
-    ]
-  ]
+intros.
+unfold sigma_p.
+apply (false_to_eq_iter_p_gen);
+  assumption.
 qed.
 
 theorem sigma_p2 : 
@@ -170,57 +112,17 @@ sigma_p (n*m)
 sigma_p n p1 
   (\lambda x.sigma_p m p2 (g x)).
 intros.
-elim n
-  [simplify.reflexivity
-  |apply (bool_elim ? (p1 n1))
-    [intro.
-     rewrite > (true_to_sigma_p_Sn ? ? ? H1).
-     simplify in \vdash (? ? (? % ? ?) ?);
-     rewrite > sigma_p_plus.
-     rewrite < H.
-     apply eq_f2
-      [apply eq_sigma_p
-        [intros.
-         rewrite > sym_plus.
-         rewrite > (div_plus_times ? ? ? H2).
-         rewrite > (mod_plus_times ? ? ? H2).
-         rewrite > H1.
-         simplify.reflexivity
-        |intros.
-         rewrite > sym_plus.
-         rewrite > (div_plus_times ? ? ? H2).
-         rewrite > (mod_plus_times ? ? ? H2).
-         rewrite > H1.
-         simplify.reflexivity.   
-        ]
-      |reflexivity
-      ]
-    |intro.
-     rewrite > (false_to_sigma_p_Sn ? ? ? H1).
-     simplify in \vdash (? ? (? % ? ?) ?);
-     rewrite > sigma_p_plus.
-     rewrite > H.
-     apply (trans_eq ? ? (O+(sigma_p n1 p1 (\lambda x:nat.sigma_p m p2 (g x)))))
-      [apply eq_f2
-        [rewrite > (eq_sigma_p ? (\lambda x.false) ? (\lambda x:nat.g ((x+n1*m)/m) ((x+n1*m)\mod m)))
-          [apply sigma_p_false
-          |intros.
-           rewrite > sym_plus.
-           rewrite > (div_plus_times ? ? ? H2).
-           rewrite > (mod_plus_times ? ? ? H2).
-           rewrite > H1.
-           simplify.reflexivity
-          |intros.reflexivity.
-          ]
-        |reflexivity
-        ]
-      |reflexivity   
-      ]
-    ]
-  ]
+unfold sigma_p.
+apply (iter_p_gen2 n m p1 p2 Z g OZ Zplus)
+[ apply symmetricZPlus
+| apply associative_Zplus
+| intros.
+  apply Zplus_z_OZ
+]
 qed.
 
 (* a stronger, dependent version, required e.g. for dirichlet product *)
+
 theorem sigma_p2' : 
 \forall n,m:nat.
 \forall p1:nat \to bool.
@@ -232,129 +134,28 @@ sigma_p (n*m)
 sigma_p n p1 
   (\lambda x.sigma_p m (p2 x) (g x)).
 intros.
-elim n
-  [simplify.reflexivity
-  |apply (bool_elim ? (p1 n1))
-    [intro.
-     rewrite > (true_to_sigma_p_Sn ? ? ? H1).
-     simplify in \vdash (? ? (? % ? ?) ?);
-     rewrite > sigma_p_plus.
-     rewrite < H.
-     apply eq_f2
-      [apply eq_sigma_p
-        [intros.
-         rewrite > sym_plus.
-         rewrite > (div_plus_times ? ? ? H2).
-         rewrite > (mod_plus_times ? ? ? H2).
-         rewrite > H1.
-         simplify.reflexivity
-        |intros.
-         rewrite > sym_plus.
-         rewrite > (div_plus_times ? ? ? H2).
-         rewrite > (mod_plus_times ? ? ? H2).
-         rewrite > H1.
-         simplify.reflexivity.   
-        ]
-      |reflexivity
-      ]
-    |intro.
-     rewrite > (false_to_sigma_p_Sn ? ? ? H1).
-     simplify in \vdash (? ? (? % ? ?) ?);
-     rewrite > sigma_p_plus.
-     rewrite > H.
-     apply (trans_eq ? ? (O+(sigma_p n1 p1 (\lambda x:nat.sigma_p m (p2 x) (g x)))))
-      [apply eq_f2
-        [rewrite > (eq_sigma_p ? (\lambda x.false) ? (\lambda x:nat.g ((x+n1*m)/m) ((x+n1*m)\mod m)))
-          [apply sigma_p_false
-          |intros.
-           rewrite > sym_plus.
-           rewrite > (div_plus_times ? ? ? H2).
-           rewrite > (mod_plus_times ? ? ? H2).
-           rewrite > H1.
-           simplify.reflexivity
-          |intros.reflexivity.
-          ]
-        |reflexivity
-        ]
-      |reflexivity   
-      ]
-    ]
-  ]
+unfold sigma_p.
+apply (iter_p_gen2' n m p1 p2 Z g OZ Zplus)
+[ apply symmetricZPlus
+| apply associative_Zplus
+| intros.
+  apply Zplus_z_OZ
+]
 qed.
 
 lemma sigma_p_gi: \forall g: nat \to Z.
 \forall n,i.\forall p:nat \to bool.i < n \to p i = true \to 
 sigma_p n p g = g i + sigma_p n (\lambda x. andb (p x) (notb (eqb x i))) g.
-intros 2.
-elim n
-  [apply False_ind.
-   apply (not_le_Sn_O i).
-   assumption
-  |apply (bool_elim ? (p n1));intro
-    [elim (le_to_or_lt_eq i n1)
-      [rewrite > true_to_sigma_p_Sn
-        [rewrite > true_to_sigma_p_Sn
-          [rewrite < assoc_Zplus.
-           rewrite < sym_Zplus in \vdash (? ? ? (? % ?)).
-           rewrite > assoc_Zplus.
-           apply eq_f2
-            [reflexivity
-            |apply H[assumption|assumption]
-            ]
-          |rewrite > H3.simplify.
-           change with (notb (eqb n1 i) = notb false).
-           apply eq_f.
-           apply not_eq_to_eqb_false.
-           unfold Not.intro.
-           apply (lt_to_not_eq ? ? H4).
-           apply sym_eq.assumption
-          ]
-        |assumption
-        ]
-      |rewrite > true_to_sigma_p_Sn
-        [rewrite > H4.
-         apply eq_f2
-          [reflexivity
-          |rewrite > false_to_sigma_p_Sn
-            [apply eq_sigma_p
-              [intros.
-               elim (p x)
-                [simplify.
-                 change with (notb false = notb (eqb x n1)).
-                 apply eq_f.
-                 apply sym_eq. 
-                 apply not_eq_to_eqb_false.
-                 apply (lt_to_not_eq ? ? H5)
-                |reflexivity
-                ]
-              |intros.reflexivity
-              ]
-            |rewrite > H3.
-             rewrite > (eq_to_eqb_true ? ? (refl_eq ? n1)).
-             reflexivity
-            ]
-          ]
-        |assumption
-        ]
-      |apply le_S_S_to_le.assumption
-      ]
-    |rewrite > false_to_sigma_p_Sn
-      [elim (le_to_or_lt_eq i n1)
-        [rewrite > false_to_sigma_p_Sn
-          [apply H[assumption|assumption]
-          |rewrite > H3.reflexivity
-          ]
-        |apply False_ind. 
-         apply not_eq_true_false.
-         rewrite < H2.
-         rewrite > H4.
-         assumption
-        |apply le_S_S_to_le.assumption
-        ]
-      |assumption
-      ]
-    ] 
-  ] 
+intros.
+unfold sigma_p.
+apply (iter_p_gen_gi)
+[ apply symmetricZPlus
+| apply associative_Zplus
+| intros.
+  apply Zplus_z_OZ
+| assumption
+| assumption
+]
 qed.
 
 theorem eq_sigma_p_gh: 
@@ -368,167 +169,22 @@ theorem eq_sigma_p_gh:
 (\forall j. j < n1 \to p2 j = true \to h (h1 j) = j) \to 
 (\forall j. j < n1 \to p2 j = true \to h1 j < n) \to 
 sigma_p n p1 (\lambda x.g(h x)) = sigma_p n1 (\lambda x.p2 x) g.
-intros 4.
-elim n
-  [generalize in match H5.
-   elim n1
-    [reflexivity
-    |apply (bool_elim ? (p2 n2));intro
-      [apply False_ind.
-       apply (not_le_Sn_O (h1 n2)).
-       apply H7
-        [apply le_n|assumption]
-      |rewrite > false_to_sigma_p_Sn
-        [apply H6.
-         intros.  
-            apply H7[apply le_S.apply H9|assumption]
-        |assumption
-        ]
-      ]
-    ]
-  |apply (bool_elim ? (p1 n1));intro
-    [rewrite > true_to_sigma_p_Sn
-      [rewrite > (sigma_p_gi g n2 (h n1))
-        [apply eq_f2
-          [reflexivity
-          |apply H
-            [intros.
-             rewrite > H1
-              [simplify.
-               change with ((\not eqb (h i) (h n1))= \not false).
-               apply eq_f.
-               apply not_eq_to_eqb_false.
-               unfold Not.intro.
-               apply (lt_to_not_eq ? ? H8).
-               rewrite < H2
-                [rewrite < (H2 n1)
-                  [apply eq_f.assumption|apply le_n|assumption]
-                |apply le_S.assumption
-                |assumption
-                ]
-              |apply le_S.assumption
-              |assumption
-              ]
-            |intros.
-             apply H2[apply le_S.assumption|assumption]
-            |intros.
-             apply H3[apply le_S.assumption|assumption]
-            |intros.
-             apply H4
-              [assumption
-              |generalize in match H9.
-               elim (p2 j)
-                [reflexivity|assumption]
-              ]
-            |intros.
-             apply H5
-              [assumption
-              |generalize in match H9.
-               elim (p2 j)
-                [reflexivity|assumption]
-              ]
-            |intros.
-             elim (le_to_or_lt_eq (h1 j) n1)
-              [assumption
-              |generalize in match H9.
-               elim (p2 j)
-                [simplify in H11.
-                 absurd (j = (h n1))
-                  [rewrite < H10.
-                   rewrite > H5
-                    [reflexivity|assumption|auto]
-                  |apply eqb_false_to_not_eq.
-                   generalize in match H11.
-                   elim (eqb j (h n1))
-                    [apply sym_eq.assumption|reflexivity]
-                  ]
-                |simplify in H11.
-                 apply False_ind.
-                 apply not_eq_true_false.
-                 apply sym_eq.assumption
-                ]
-              |apply le_S_S_to_le.
-               apply H6
-                [assumption
-                |generalize in match H9.
-                 elim (p2 j)
-                  [reflexivity|assumption]
-                ]
-              ]
-            ]
-          ]
-        |apply H3[apply le_n|assumption]
-        |apply H1[apply le_n|assumption]
-        ]
-      |assumption
-      ]
-    |rewrite > false_to_sigma_p_Sn
-     [apply H
-       [intros.apply H1[apply le_S.assumption|assumption]
-       |intros.apply H2[apply le_S.assumption|assumption]
-       |intros.apply H3[apply le_S.assumption|assumption]
-       |intros.apply H4[assumption|assumption]
-       |intros.apply H5[assumption|assumption]
-       |intros.
-        elim (le_to_or_lt_eq (h1 j) n1)
-          [assumption
-          |absurd (j = (h n1))
-            [rewrite < H10.
-             rewrite > H5
-               [reflexivity|assumption|assumption]
-            |unfold Not.intro.
-             apply not_eq_true_false.
-             rewrite < H7.
-             rewrite < H10.
-             rewrite > H4
-              [reflexivity|assumption|assumption]
-            ]
-          |apply le_S_S_to_le.
-           apply H6[assumption|assumption]
-          ]
-        ]
-      |assumption
-      ]
-    ]
-  ]
-qed.
-
-(* sigma_p and Ztimes *)
-lemma Ztimes_sigma_pl: \forall z:Z.\forall n:nat.\forall p. \forall f.
-z * (sigma_p n p f) = sigma_p n p (\lambda i.z*(f i)).
 intros.
-elim n
-  [rewrite > Ztimes_z_OZ.reflexivity
-  |apply (bool_elim ? (p n1)); intro
-    [rewrite > true_to_sigma_p_Sn
-      [rewrite > true_to_sigma_p_Sn 
-        [rewrite < H.
-         apply distr_Ztimes_Zplus
-        |assumption
-        ]
-      |assumption
-      ]
-    |rewrite > false_to_sigma_p_Sn
-      [rewrite > false_to_sigma_p_Sn
-        [assumption
-        |assumption
-        ]
-      |assumption
-      ] 
-    ]
-  ]
+unfold sigma_p.
+apply (eq_iter_p_gen_gh Z OZ Zplus ? ? ? g h h1 n n1 p1 p2)
+[ apply symmetricZPlus
+| apply associative_Zplus
+| intros.
+  apply Zplus_z_OZ
+| assumption
+| assumption
+| assumption
+| assumption
+| assumption
+| assumption
+]
 qed.
 
-lemma Ztimes_sigma_pr: \forall z:Z.\forall n:nat.\forall p. \forall f.
-(sigma_p n p f) * z = sigma_p n p (\lambda i.(f i)*z).
-intros.
-rewrite < sym_Ztimes.
-rewrite > Ztimes_sigma_pl.
-apply eq_sigma_p
-  [intros.reflexivity
-  |intros.apply sym_Ztimes
-  ]
-qed.
 
 theorem divides_exp_to_lt_ord:\forall n,m,j,p. O < n \to prime p \to
 p \ndivides n \to j \divides n*(exp p m) \to ord j p < S m.
@@ -608,6 +264,7 @@ cut (O < j)
   ] 
 qed.
 
+
 theorem sigma_p_divides_b: 
 \forall n,m,p:nat.O < n \to prime p \to Not (divides p n) \to 
 \forall g: nat \to Z.
@@ -615,153 +272,43 @@ sigma_p (S (n*(exp p m))) (\lambda x.divides_b x (n*(exp p m))) g =
 sigma_p (S n) (\lambda x.divides_b x n)
   (\lambda x.sigma_p (S m) (\lambda y.true) (\lambda y.g (x*(exp p y)))).
 intros.
-cut (O < p)
-  [rewrite < sigma_p2.
-   apply (trans_eq ? ? 
-    (sigma_p (S n*S m) (\lambda x:nat.divides_b (x/S m) n)
-       (\lambda x:nat.g (x/S m*(p)\sup(x\mod S m)))))
-    [apply sym_eq.
-     apply (eq_sigma_p_gh g ? (p_ord_inv p (S m)))
-      [intros.
-       lapply (divides_b_true_to_lt_O ? ? H H4).
-       apply divides_to_divides_b_true
-        [rewrite > (times_n_O O).
-         apply lt_times
-          [assumption
-          |apply lt_O_exp.assumption
-          ]
-        |apply divides_times
-          [apply divides_b_true_to_divides.assumption
-          |apply (witness ? ? (p \sup (m-i \mod (S m)))).
-           rewrite < exp_plus_times.
-           apply eq_f.
-           rewrite > sym_plus.
-           apply plus_minus_m_m.
-           auto
-          ]
-        ]
-      |intros.
-       lapply (divides_b_true_to_lt_O ? ? H H4).
-       unfold p_ord_inv.
-       rewrite > (p_ord_exp1 p ? (i \mod (S m)) (i/S m))
-        [change with ((i/S m)*S m+i \mod S m=i).
-         apply sym_eq.
-         apply div_mod.
-         apply lt_O_S
-        |assumption
-        |unfold Not.intro.
-         apply H2.
-         apply (trans_divides ? (i/ S m))
-          [assumption|
-           apply divides_b_true_to_divides;assumption]
-        |apply sym_times.
-        ]
-      |intros.
-       apply le_S_S.
-       apply le_times
-        [apply le_S_S_to_le.
-         change with ((i/S m) < S n).
-         apply (lt_times_to_lt_l m).
-         apply (le_to_lt_to_lt ? i)
-          [auto|assumption]
-        |apply le_exp
-          [assumption
-          |apply le_S_S_to_le.
-           apply lt_mod_m_m.
-           apply lt_O_S
-          ]
-        ]
-      |intros.
-       cut (ord j p < S m)
-        [rewrite > div_p_ord_inv
-          [apply divides_to_divides_b_true
-            [apply lt_O_ord_rem
-              [elim H1.assumption
-              |apply (divides_b_true_to_lt_O ? ? ? H4).
-                rewrite > (times_n_O O).
-                apply lt_times
-                [assumption|apply lt_O_exp.assumption]
-              ]
-            |apply (divides_exp_to_divides_ord_rem ? m ? ? H H1 H2).
-             apply divides_b_true_to_divides.
-             assumption
-            ]
-            |assumption
-          ]
-        |apply (divides_exp_to_lt_ord ? ? ? ? H H1 H2).
-         apply (divides_b_true_to_divides ? ? H4).
-         apply (divides_b_true_to_lt_O ? ? H4)
-        ]
-      |intros.
-       cut (ord j p < S m)
-        [rewrite > div_p_ord_inv
-          [rewrite > mod_p_ord_inv
-            [rewrite > sym_times.
-             apply sym_eq.
-             apply exp_ord
-              [elim H1.assumption
-              |apply (divides_b_true_to_lt_O ? ? ? H4).
-               rewrite > (times_n_O O).
-               apply lt_times
-                [assumption|apply lt_O_exp.assumption]
-              ]
-            |apply (divides_exp_to_lt_ord ? ? ? ? H H1 H2).
-             apply (divides_b_true_to_divides ? ? H4).
-             apply (divides_b_true_to_lt_O ? ? H4)
-            ]
-          |assumption
-          ]
-        |apply (divides_exp_to_lt_ord ? ? ? ? H H1 H2).
-         apply (divides_b_true_to_divides ? ? H4).
-         apply (divides_b_true_to_lt_O ? ? H4).
-        ]
-      |intros.
-       rewrite > eq_p_ord_inv.
-       rewrite > sym_plus.
-       apply (lt_to_le_to_lt ? (S m +ord_rem j p*S m))
-        [apply lt_plus_l.
-         apply le_S_S.
-         cut (m = ord (n*(p \sup m)) p)
-          [rewrite > Hcut1.
-           apply divides_to_le_ord
-            [apply (divides_b_true_to_lt_O ? ? ? H4).
-             rewrite > (times_n_O O).
-             apply lt_times
-              [assumption|apply lt_O_exp.assumption]
-            |rewrite > (times_n_O O).
-             apply lt_times
-              [assumption|apply lt_O_exp.assumption]
-            |assumption
-            |apply divides_b_true_to_divides.
-             assumption
-            ]
-          |unfold ord.
-           rewrite > sym_times.
-           rewrite > (p_ord_exp1 p ? m n)
-            [reflexivity
-            |assumption
-            |assumption
-            |reflexivity
-            ]
-          ]
-        |change with (S (ord_rem j p)*S m \le S n*S m).
-         apply le_times_l.
-         apply le_S_S.
-         apply divides_to_le
-          [assumption
-          |apply (divides_exp_to_divides_ord_rem ? m ? ? H H1 H2).
-           apply divides_b_true_to_divides.
-           assumption
-          ]
-        ]
-      ]
-    |apply eq_sigma_p
-      [intros.
-       elim (divides_b (x/S m) n);reflexivity
-      |intros.reflexivity
-      ]
-    ]
-  |elim H1.apply lt_to_le.assumption
-  ]
+unfold sigma_p.
+apply (iter_p_gen_divides Z OZ Zplus n m p ? ? ? g)
+[ assumption
+| assumption
+| assumption
+| apply symmetricZPlus
+| apply associative_Zplus
+| intros.
+  apply Zplus_z_OZ
+]
 qed.
+
     
+(* sigma_p and Ztimes *)
+lemma Ztimes_sigma_pl: \forall z:Z.\forall n:nat.\forall p. \forall f.
+z * (sigma_p n p f) = sigma_p n p (\lambda i.z*(f i)).
+intros.
+apply (distributive_times_plus_iter_p_gen Z Zplus OZ Ztimes n z p f)
+[ apply symmetricZPlus
+| apply associative_Zplus
+| intros.
+  apply Zplus_z_OZ
+| apply symmetric_Ztimes
+| apply distributive_Ztimes_Zplus
+| intros.
+  rewrite > (Ztimes_z_OZ a).
+  reflexivity
+]
+qed.
+
+lemma Ztimes_sigma_pr: \forall z:Z.\forall n:nat.\forall p. \forall f.
+(sigma_p n p f) * z = sigma_p n p (\lambda i.(f i)*z).
+intros.
+rewrite < sym_Ztimes.
+rewrite > Ztimes_sigma_pl.
+apply eq_sigma_p
+  [intros.reflexivity
+  |intros.apply sym_Ztimes
+  ]
+qed.